Michael selection theorem

{{short description|On the existence of a continuous selection of a multivalued map from a paracompact space}}

In functional analysis, a branch of mathematics, Michael selection theorem is a selection theorem named after Ernest Michael. In its most popular form, it states the following:{{Cite journal | last1=Michael | first1=Ernest | author1-link=Ernest Michael | title=Continuous selections. I |mr=0077107 | year=1956 | journal=Annals of Mathematics |series=Second Series | volume=63 | pages=361–382 | issue=2 | jstor=1969615 | doi=10.2307/1969615| hdl=10338.dmlcz/119700 | hdl-access=free }}

{{Math theorem

| name = Michael Selection Theorem|Let X be a paracompact space and Y be a separable Banach space.

Let $F\colon X\to Y$ be a lower hemicontinuous set-valued function with nonempty convex closed values.

Then there exists a continuous selection $f\colon X \to Y$ of F.

}}

Conversely, if any lower semicontinuous multimap from topological space X to a Banach space, with nonempty convex closed values, admits a continuous selection, then X is paracompact. This provides another characterization for paracompactness.

Examples

File:Kakutani.svg

= A function that satisfies all requirements =

The function: $F(x)=
[1-x/2, ~1-x/4]$ , shown by the grey area in the figure at the right, is a set-valued function from the real interval [0,1] to itself. It satisfies all Michael's conditions, and indeed it has a continuous selection, for example: $f(x)=
1-x/2$ or $f(x)=
1-3x/8$ .{{clear}}

= A function that does not satisfy lower hemicontinuity =

The function

$F(x)=
\begin{cases}
3/4 & 0 \le x < 0.5 \\
\left[0,1\right] & x = 0.5 \\
1/4 & 0.5 < x \le 1
\end{cases}$

is a set-valued function from the real interval [0,1] to itself. It has nonempty convex closed values. However, it is not lower hemicontinuous at 0.5. Indeed, Michael's theorem does not apply and the function does not have a continuous selection: any selection at 0.5 is necessarily discontinuous.{{Cite web|url=https://math.stackexchange.com/q/3377063 |title=proof verification - Reducing Kakutani's fixed-point theorem to Brouwer's using a selection theorem|website=Mathematics Stack Exchange|access-date=2019-10-29}}

Applications

Michael selection theorem can be applied to show that the differential inclusion

: $\frac{dx}{dt}(t)\in F(t,x(t)), \quad x(t_0)=x_0$

has a C¹ solution when F is lower semi-continuous and F(t, x) is a nonempty closed and convex set for all (t, x). When F is single valued, this is the classic Peano existence theorem.

Generalizations

A theorem due to Deutsch and Kenderov generalizes Michel selection theorem to an equivalence relating approximate selections to almost lower hemicontinuity, where $F$ is said to be almost lower hemicontinuous if at each $x \in X$ , all neighborhoods $V$ of $0$ there exists a neighborhood $U$ of $x$ such that $\cap_{u\in U} \{F(u)+V\} \ne \emptyset.$

Precisely, Deutsch–Kenderov theorem states that if $X$ is paracompact, $Y$ a normed vector space and $F (x)$ is nonempty convex for each $x \in X$ , then $F$ is almost lower hemicontinuous if and only if $F$ has continuous approximate selections, that is, for each neighborhood $V$ of $0$ in $Y$ there is a continuous function $f \colon X \mapsto Y$ such that for each $x \in X$ , $f (x) \in F (X) + V$ .{{cite journal|last1=Deutsch|first1=Frank|last2=Kenderov|first2=Petar|title=Continuous Selections and Approximate Selection for Set-Valued Mappings and Applications to Metric Projections|journal=SIAM Journal on Mathematical Analysis|date=January 1983|volume=14|issue=1|pages=185–194|doi=10.1137/0514015}}

In a note Xu proved that Deutsch–Kenderov theorem is also valid if $Y$ is a locally convex topological vector space.{{cite journal|last1=Xu|first1=Yuguang|title=A Note on a Continuous Approximate Selection Theorem|journal=Journal of Approximation Theory|date=December 2001|volume=113|issue=2|pages=324–325|doi=10.1006/jath.2001.3622|doi-access=free}}

Michael selection theorem

Examples

= A function that satisfies all requirements =

= A function that does not satisfy lower hemicontinuity =

Applications

Generalizations

See also

References

Further reading